Solve two-equation systems by substitution or elimination, and decide when a system has no solution.
You solve two linear equations in two unknowns, usually by substitution or elimination. Questions may ask for one variable, for both, or for a combined expression such as the sum of the two variables. Word problems set up systems from two constraints, most often a count constraint and a total-value constraint. A second family of questions asks about the number of solutions: a system has no solution when the lines are parallel, and infinitely many when the two equations describe the same line.
Systems show up often, usually around 2 to 4 of the 44 math questions, since Algebra is roughly 35 percent of the section.
Written for this skill, with the answer explained. 170 more are in the app.
The graphs of the two linear equations in a system are shown as lines ℓ and k in the xy-plane. What is the solution (x, y) to the system?
The graph shows the total charge, in dollars, for renting a van for m miles from two companies. What is the total charge, in dollars, at the mileage where the two companies charge the same amount?
Why: The charges are equal where the lines cross. Company B starts 20 dollars lower but gains 0.50 per mile on Company A, so they meet after 20/0.5 = 40 miles, where the charge is 0.5(40) + 40 = 60 dollars. The 40 is the mileage, not the charge.
For what value of k does kx + 2y = 6 and 3x + y = 4 have no solution?
Why: Write both in slope form: kx + 2y = 6 has slope -k/2 and 3x + y = 4 has slope -3. No solution needs parallel lines, so -k/2 = -3 and k = 6, and the intercepts 3 and 4 differ, so the lines really are parallel. Answering 3 copies the other equation's x-coefficient, which gives slope -3/2. Answering 2 uses its y-coefficient, giving slope -1, and 1/3 gives slope -1/6, so each of those lines still crosses the second line.
A dye shop mixes a 6% bleach solution with a 21% bleach solution. It needs 45 liters of a 16% solution. How many liters of the 6% solution are needed?
Why: The target holds 0.16(45) = 7.2 liters of bleach, and 45 liters of the 6% solution would hold 2.7. Each liter of the 21% solution adds 0.15 liter, so 4.5/0.15 = 30 liters of the 21% solution and 15 liters of the 6% are needed. Answering 30 reports the wrong ingredient. Answering 4.5 means you reported the 4.5 liters of extra bleach rather than a volume of solution. Answering 22.5 means you assumed equal volumes of the two solutions. Answering 30 means you found the liters of the 21% solution instead of the 6%.
Use substitution when one equation is already solved for a variable or has a coefficient of 1, since isolating is free. Use elimination when both equations are in standard form with matching variable layouts. Elimination is usually faster when the question asks for a combination such as x plus y.
When the two lines have the same slope but different y-intercepts, so they are parallel and never meet. Algebraically, one equation is a multiple of the other on the variable side but not on the constant side. If both sides scale by the same factor, there are infinitely many solutions instead.
Name each unknown with a letter and write the units next to it. Most SAT setups give you one equation counting items and a second equation totaling money, weight, or time. Write both before you solve anything.
Questions on systems of linear equations alone, with explanations on every one, and the app tracks which ones you keep missing.
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